Effective Spring Constant of a Chain Polymer

[omega_minus]

TL;DR Summary

Looking for resources to help model the effective spring constant of chain molecules sorbed onto the surface of magnetic nanoparticles

Hi All,

I'm doing research in magnetic nanoparticles that are coated with chain molecules (oleic acid, I believe) and I am trying to model these molecules' effective spring constant.
The basic scenario is this: When a water-based ferrofluid is evaporated, it leaves behind only dried particles that are clumped together. These particles have chain molecules sorbed onto their surface to prevent agglomeration and oxidization while in the carrier fluid. The protective molecules are about 1nm long. One they are dried, if they are subjected to a magnetic field they will try to rotate to align in the field. To know how they will do this a priori, I'd like to model them as simple springs. I have looked in several places online but I can't find where this property is discussed (or at least it's not called a spring constant). With this info I could perhaps make a more sophisticated model later.
Some extra details for anyone interested: The particles are 10nm Fe3O4 and have an internal anisotropy around 25-30kA/m based on fitted data and a theoretical calculation for cubic anisotropy. I know the magnetization will align in the field independent of the particle crystal, but it won't be without some strain on the bonds that connect each particle due to the non-zero anisotropy.
Any help is appreciated. I have access to the Web of Knowledge through the graduate school so even a DOI or paper title will suffice.

Thanks

 

[klotza]

The model you are looking for is the Ideal Chain. It is based on a chain of monomers that can rotate freely (or in one dimension, be aligned or anti-aligned with their neighbors) and ignores interactions between them. You can essentially treat the location of the Nth monomer like a Brownian particle that has gone N time-steps, such that its mean distance from the start of the chain (averaged over every possible configuration) is zero, but its mean square distance is nonzero. Just like the mean square distance traveled by a Brownian particle is proportional to time (the proportionality being the diffusion coefficient), the mean square distance between two ends of a polymer chain proportional to the number of links in the chain (N) and the proportionality is the squared length of each link (b), such that <X>=b√N. When N is large <X> will be much less than the total length bN.

The spring constant can be found by taking the end position of the chain to be Gaussianly distributed around the start position (again, like the location of a random walker), which when you apply the Boltzmann distribution (relating the energy of a state in contact with a thermal reservoir to the probability of observing that state), you find the Gaussian probability becomes a quadratic energy, which of course is what describes a spring system. Because the force is entropic (the chains contract to increase entropy), the energy scale of the quadratic potential is kT (Boltzmann constant times temperature), and for the dimensions to make sense we want a spring force described by f=Kx=kT x/C, where C has to have units of length squared for the whole equation to have units of force. We can handwave that C should be some kind of product of the link size b, and the total length L=Nb.

The actual answer is that f=-3 kT x/(Lb) so the spring constant is 3kT/(Lb).

See here: https://en.wikipedia.org/wiki/Ideal_chain

 

[omega_minus]

The model you are looking for is the Ideal Chain. It is based on a chain of monomers that can rotate freely (or in one dimension, be aligned or anti-aligned with their neighbors) and ignores interactions between them. You can essentially treat the location of the Nth monomer like a Brownian particle that has gone N time-steps, such that its mean distance from the start of the chain (averaged over every possible configuration) is zero, but its mean square distance is nonzero. Just like the mean square distance traveled by a Brownian particle is proportional to time (the proportionality being the diffusion coefficient), the mean square distance between two ends of a polymer chain proportional to the number of links in the chain (N) and the proportionality is the squared length of each link (b), such that <X>=b√N. When N is large <X> will be much less than the total length bN.

The spring constant can be found by taking the end position of the chain to be Gaussianly distributed around the start position (again, like the location of a random walker), which when you apply the Boltzmann distribution (relating the energy of a state in contact with a thermal reservoir to the probability of observing that state), you find the Gaussian probability becomes a quadratic energy, which of course is what describes a spring system. Because the force is entropic (the chains contract to increase entropy), the energy scale of the quadratic potential is kT (Boltzmann constant times temperature), and for the dimensions to make sense we want a spring force described by f=Kx=kT x/C, where C has to have units of length squared for the whole equation to have units of force. We can handwave that C should be some kind of product of the link size b, and the total length L=Nb.

The actual answer is that f=-3 kT x/(Lb) so the spring constant is 3kT/(Lb).

See here: https://en.wikipedia.org/wiki/Ideal_chain

@klotza Thank you for such a detailed response! This is exactly what I was looking for. As an electrical engineer my knowledge of these kinds of things is somewhat limited but now I can see what you are saying. It will take a while to digest but this has set me on the right path. Thanks again for everything